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euclid.jap.1261670699

euclid.jap.1261670699 - J Appl Prob 46 12091212(2009...

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J. Appl. Prob. 46, 1209–1212 (2009) Printed in England © Applied Probability Trust 2009 AN INEQUALITY FOR VARIANCES OF THE DISCOUNTED REWARDS EUGENE A. FEINBERG ∗ ∗∗ and JUN FEI, ∗ ∗∗∗ Stony Brook University Abstract We consider the following two definitions of discounting: (i) multiplicative coefficient in front of the rewards, and (ii) probability that the process has not been stopped if the stopping time has an exponential distribution independent of the process. It is well known that the expected total discounted rewards corresponding to these definitions are the same. In this note we show that, the variance of the total discounted rewards is smaller for the first definition than for the second definition. Keywords: Total discounted reward; variance; stopping time 2000 Mathematics Subject Classification: Primary 60G40 Secondary 90C40 1. Introduction In this note we study two definitions of discounting: (i) multiplicative coefficient in front of the rewards, and (ii) probability that the process has not been stopped if the stopping time has an exponential distribution independent of the process. It is well known that the total discounted rewards corresponding to these definitions have equal expectations. However, as we will show, the second moment and variance are smaller for the first definition than for the second definition. Since its introduction by Markowitz in his Nobel Prize winning paper [5], variance has played an important role in stochastic optimization. In particular, there is a significant amount of literature on various optimizations of Markov decision processes (MDPs); see the pioneering work by Jaquette [4] and Sobel [7]–[9], a survey by White [11], and recent references by Van Dijk and Sladký [10] and Baykal-Gürsoy and Gürsoy [1].
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